MATH 1117 Yale University Functions as A Polynomial Midterm Exam ………………………………………………………………………… MATH1

MATH 1117 Yale University Functions as A Polynomial Midterm Exam ………………………………………………………………………… MATH1117 Practice Questions for the Midterm
Instructions: This document contains the types of problems that you should be prepared to answer
for the upcoming midterm exam. There are many more questions here than will be on the actual midterm.
Solutions to these questions are not provided. If you do want to gauge your understanding of these
questions, you are encouraged to attend the office hours of the instructor or learning assistant and to ask
specific questions about your work. In addition to these questions, the homework and worksheets are good
study materials.
1. Classify the following functions as a polynomial, rational function, or algebraic function. Choose all
that apply.
(a) f (x) = √
3x + 2
(b) g(x) = 4x
3x + 4
(c) h(t) = 4
3x − 2×2 + 1
1
2
2. Which of the following functions have inverses?
3
3. Find the inverse of the following functions
(a) f (x) = 4x + 2
(b) g(t) = (x − 2)2 + 3 on the domain x ≥ 2
(c) g(t) = (x − 2)2 + 3 on the domain x ≤ 2
ln(3x − 2)
(d) h(x) =
4
(e) f (t) = 4 sin(x − π)
4
4. Combine the given functions as requested. Do not simplify your answers.
(a) What is (f /g)(x) given f (x) = 3×2 and g(x) = sin(2x)
(b) What is (f + g)(x) given f (x) = tan(2x − 3) and g(x) = e3x
(c) What is (f ◦ g)(x) for f (x) = cos(x) and g(x) = 3×2
(d) What is (f ◦ g)(x) for f (x) = e2x and g(x) = x2 − 2x + 1
(e) What is (f ◦ g)(x) for f (x) = x2 + 2x + 1 and g(x) = 3×2 − 2x − 1
5
5. For each of the following graphs, find lim f (x), lim f (x), lim f (x)
x→0−
x→0+
x→0
6
6. Compute the following limits.
(a) lim 3×2 − 3x + 1
x→−2
x2 − 4x + 4
x→0
x+1
x3 − 9x
lim
x→3 x − 3
x−4
lim √
x→4
x−2
x2 + x − 2
lim 2
x→1 (
x + 2x − 3
2x − 2, x < 4 lim x→4 4x − 9, x ≥ 4 1 lim x2 sin x→0 x 1 lim x2 +4x+4 (b) lim (c) (d) (e) (f) (g) (h) x→0 x 2 x→1 x +x−2 lim 5x2 − 15x + 10 x→−∞ x5 + 2x2 + 17 (i) lim (j) (k) lim 4x2 + 1 18x7 − 2x + 1 (l) lim x→∞ 3x7 − 34x + 7 14x4 + 7x3 − 3x2 (m) lim x→−∞ 4x9 − 23x4 + 2x x→∞ 7 7. State whether the given functions are continuous at the specified point x2 + 2x + 1 (a) f (x) = at x = 4 ( x−4 x2 , x ≥ 0 (b) f (x) = , at x = 0 1, x Purchase answer to see full attachment